Chapter 9The diffuse structure of the electrical double layer

When considering electrosmotic flows, Chapter 6 focuses on outer solutions, namely solutions for flow far from
boundaries. In this limit, we describe electroosmosis using an effectiveslip boundary condition wall= μEO. A 1D
integral model of the surface shows that, if the fluid properties are assumed uniform and the electrical potential at the
wall is different from the bulk by a factor of φ0, then μEO is given by μEO= -εφ0∕η. The inner distribution of
velocity and electrical potential need not be determined.

In this chapter, we address the electrical double layer (also called the Debye layer) near a charged wall and
evaluate the spatial variation of charge and potential in this double layer. This determines the equilibrium structure
of the fluid boundary layer near a surface in an electroosmotically driven system and describes the spatial
variation of velocity near the wall. In the process, we relate the Coulomb force (related to the total wall
charge density) and the distribution of the Coulomb force (related to the Debye length) to the velocity
distribution. The flows that result are the inner solutions of electroosmotic flow problems. In total, the
equilibrium electrical double layer solution leads to predictions of fluid flow and current in electrically driven
micro- and nanofluidic systems that do not perturb this equilibrium. As this chapter involves detailed
discussion of ion concentrations, a review of the terminology and parameters found in Appendix B is
recommended.